First we will begin making the dough. Add water to a large bowl, along with 3½ cups of the flour, 2 tablespoons of oil, salt, sugar, egg, and yeast. Stir together with a wooden spoon until combined. The dough will be very thick but still pretty wet and sticky.
Place 2 tablespoons of flour on the surface of your counter. Place dough on top of it and begin kneading the dough with the flour until all the flour is mixed into the dough. Keep kneading for about 4-5 minutes until the dough is smooth and elastic. If it starts to stick to the counter, add a little more flour. Once you finish kneading, shape into a ball. Lightly grease another large bowl and place the ball of dough into bowl. Cover with a kitchen towel and let it rise for 1 hours, where it will be double the size. I like to place mine in the microwave to keep it away from any cool draft from the A/C.
Prepare a 13x9 rectangular pan by spraying with non-stick spray. Once dough has risen, punch it down with your fist and remove from the bowl. Lightly dust your counter with flour and place the dough on top. Roll out dough into a large rectangle, 18 inches x 22 inches. Spread pesto over the dough in one even layer. Sprinkle with 1½ cups of mozzarella cheese in an even layer. Top with sprinkles of Parmesan cheese.
Starting on the long edge, tightly roll the dough into a log. Cut the edges off and then cut the log in half and then into quarters so you have 4 parts. Cut each part into 3 even rolls. Place into prepared baking pan into a 4x3 pattern. Cover and let them rise for another 30 minutes or until doubled in size. Preheat oven to 350 degrees while the rolls rise.
Once they're done rising, uncover them, sprinkle with ½ cup of more cheese, and place into the oven to bake for 25 minutes or until they've browned on top. Remove from oven and let them cool for 15 minutes before serving.